師資
獲西北工業(yè)大學(xué)飛機(jī)結(jié)構(gòu)強(qiáng)度專業(yè)本科和碩士學(xué)位,，后留校任教,。1987年赴美國(guó)留學(xué),，1992年在美國(guó)伊利諾大學(xué)香檳分校(UIUC)獲理論與應(yīng)用力學(xué)博士學(xué)位,。1992-1994年在愛(ài)荷華州立大學(xué)(ISU)做博士后研究,，1994-1996年先后在美國(guó)MSC.Software, Weidlinger Associates 和Ford Motor Company做工程師,。1996年起在美國(guó)辛辛那提大學(xué)(UC)機(jī)械工程系任職,，歷任助理教授,、副教授,、教授,、CAE研究室主任及P&G-UC高性能計(jì)算研究中心技術(shù)主任。研究方向?yàn)橛?jì)算力學(xué)及其在模擬先進(jìn)材料,、結(jié)構(gòu)動(dòng)力和聲場(chǎng)中的應(yīng)用,。近年來(lái)專注于快速邊界元方法在大規(guī)模位勢(shì)、彈性力學(xué),、流體力學(xué)和聲學(xué)中的算法研究,。英文《快速多極邊界元方法》一書(shū)2009年由劍橋大學(xué)出版社出版，在國(guó)際期刊和學(xué)術(shù)會(huì)議文集中發(fā)表論文100余篇。現(xiàn)為ASME Fellow (美國(guó)機(jī)械工程師學(xué)會(huì)會(huì)士）,，國(guó)際邊界元協(xié)會(huì) (IABEM) 執(zhí)行理事,，國(guó)際雜志Computers & Structures 及Engineering Analysis with Boundary Elements編委。獲2002年和2013年辛辛那提大學(xué)工程學(xué)院教師杰出研究獎(jiǎng),；2004年日本JSPS高級(jí)研究學(xué)者獎(jiǎng)勵(lì) (在京都大學(xué)做訪問(wèn)教授) ,；2004年中國(guó)教育部春暉計(jì)劃獎(jiǎng)勵(lì) (在清華大學(xué)做訪問(wèn)教授) ；和2012年中國(guó)杜慶華工程計(jì)算方法獎(jiǎng),。
學(xué)習(xí)經(jīng)歷
1992 美國(guó)伊利諾大學(xué)香檳分校（UIUC）理論與應(yīng)用力學(xué)（TAM）博士學(xué)位
1984 西北工業(yè)大學(xué)飛機(jī)系固體力學(xué)碩士學(xué)位
1982 西北工業(yè)大學(xué)飛機(jī)系飛機(jī)結(jié)構(gòu)強(qiáng)度學(xué)士學(xué)位
工作經(jīng)歷
2018-今 南方科技大學(xué)力學(xué)與航空航天工程系講席教授
1996-2018 美國(guó)辛辛那提大學(xué)機(jī)械工程系助理教授,、 副教授、教授
1995-1996 美國(guó)福特汽車公司先進(jìn)車輛技術(shù)部計(jì)算機(jī)輔助工程（CAE）分析師
1995 美國(guó)Weidlinger Associates公司結(jié)構(gòu)分析工程師
1994 美國(guó)MSC/Software公司軟件工程師
1992-1994 美國(guó)愛(ài)荷華州立大學(xué)無(wú)損傷探測(cè)研究中心（CNDE）博士后
1984-1987 西北工業(yè)大學(xué)飛機(jī)系飛機(jī)結(jié)構(gòu)強(qiáng)度教研室教師
榮譽(yù)及獲獎(jiǎng)
美國(guó)機(jī)械工程師學(xué)會(huì)會(huì)士（ASME Fellow, 2015）
中國(guó)杜慶華工程計(jì)算方法獎(jiǎng)（2012）
美國(guó)辛辛那提大學(xué)工學(xué)院杰出教師研究獎(jiǎng)（2002,，2013）
日本科學(xué)促進(jìn)會(huì)（JSPS）高級(jí)研究員學(xué)者獎(jiǎng)勵(lì)（2004）
中國(guó)教育部春暉計(jì)劃獎(jiǎng)勵(lì)（2004）
研究方向
計(jì)算固體力學(xué),、斷裂力學(xué)、結(jié)構(gòu)振動(dòng)及聲學(xué)
快速邊界元方法（BEM）,、有限元方法（FEM）及無(wú)網(wǎng)格方法（MFS,、MD、PD）
復(fù)合材料,、納米材料,、生物材料、聲學(xué)超材料性能多尺度數(shù)值模擬
CAE軟件開(kāi)發(fā)與應(yīng)用（FastBEM Acoustics）
近期科研項(xiàng)目：
多裂紋擴(kuò)展快速邊界元模擬及其在頁(yè)巖油氣水壓致裂開(kāi)采中的應(yīng)用,，中國(guó)國(guó)家自然科學(xué)基金,，常規(guī)面上項(xiàng)目，75萬(wàn)元,，1/2015-12/2018,，PI。
Workshop on the Boundary Element Method (BEM): Bridging Education and Industrial Applications; Minneapolis, Minnesota; 23-26 April 2012, National Science Foundation, $87,508, 10/1/11-9/30/13 (Co-PI).
SNM GOALI: Carbon Nanotube Superfiber to Revolutionize Engineering Designs, National Science Foundation, $1.8M, 9/15/11-8/31/16 (Co-PI).
High-Fidelity Prediction of Launch Vehicle Liftoff Acoustic Fields, NASA STTR Phase I (CFDRC), $30,000, 2/18/11-2/18/12 (PI).
NSF Workshop on the Emerging Applications and Future Directions of the Boundary Element Method, National Science Foundation, $31,170, 5/15/10-4/30/11 (PI).
UC High-Performance Modeling and Simulation Center, Procter & Gamble (P&G), $1.5M, starting 9/1/08 (Co-PI, 50% of efforts).
Application Engineers for TechnoSoft, TechnoSoft, $141K, 1/1/10-12/31/10 (Co-PI, 33.3% of efforts).
Collaborative Research: Fast Simulation of Complex MEMS Structures with the Boundary Element, Finite Element and Fast Multipole Methods, National Science Foundation (CMMI 0508232), $125K, 9/1/05-8/31/09 (PI).
REU – Collaborative Research: Fast Simulation of Complex MEMS Structures with the Boundary Element, Finite Element and Fast Multipole Methods, National Science Foundation (CMMI 0722767), $5.5K, 6/1/07-8/31/07 (PI).
Material Characterization for Hypersonic Vehicles by the Fast Multipole Boundary Element Method, NASA SBIR Phase I (Avant Analysis Technology), $23.3K, 1/19/07-7/18/07 (PI).
An Integrated Testing and Simulation Lab for Undergraduate Education in Mechanical Engineering, P&G Higher Education Grant, $10K, 2008 (PI).
Three-Dimensional Modeling of the Interphases in Fiber-Reinforced Composite Materials, National Science Foundation (CMS 9734949), $113K, 4/15/98-3/31/01 (PI).
學(xué)術(shù)成果
S. Huang and Y. J. Liu, “A new fast direct solver for the boundary element method,” Computational Mechanics, 60, No. 3, 379–392 (2017).
G. Hou, D. Chauhan, V. Ng, C. Xu, Z. Yin, M. Paine, R. Su, V. Shanov, D. Mast, M. Schulz, and Y. J. Liu, “Gas phase pyrolysis synthesis of carbon nanotubes at high temperature,” Materials & Design, 132, 112-118 (2017).
G. Hou, V. Ng, C. Xu, L. Zhang, G. Zhang, V. Shanov, D. Mast, W. Kim, M. Schulz, and Y. J. Liu, “Multiscale modeling of carbon nanotube bundle agglomeration inside a gas phase pyrolysis reactor,” MRS Advances, 1-6 (5/2017).
G. Hou, V. Ng, Y. Song, L. Zhang, C. Xu, V. Shanov, D. Mast, M. Schulz, and Y. J. Liu, “Numerical and experimental investigation of carbon nanotube sock formation,” MRS Advances, 2(1), 21-26 (2017).
Y. J. Liu, Y. X. Li, and W. Xie, “Modeling of multiple crack propagation in 2-D elastic solids by the fast multipole boundary element method,” Engineering Fracture Mechanics, 172, 1-16 (2017).
Y. J. Liu, “On the displacement discontinuity method and the boundary element method for solving 3-D crack problems,” Engineering Fracture Mechanics, 164, 35-45 (2016).
S. Huang and Y. J. Liu, “A new simple multidomain fast multipole boundary element method,” Computational Mechanics, 58, No. 3, 533–548 (2016).
G. Hou, R. Su, A. Wang, V. Ng, W. Li, Y. Song, L. Zhang, M. Sundaram, V. Shanov, D. Mast, D. Lashmore, M. Schulz, and Y. J. Liu, “The effect of a convection vortex on sock formation in the floating catalyst method for carbon nanotube synthesis,” Carbon, 102, 513-519 (2016).
Y. Sun, Y. J. Liu, and F. Xu, “ReaxFF molecular dynamics study on oxidation behavior of 3C-SiC: Polar face effects,” Chinese Physics B, 24, 096203 (2015).
Y. C. Cao, L. H. Wen, J. Y. Xiao, and Y. J. Liu, “A fast directional BEM for large-scale acoustic problems based on the Burton-Miller formulation,” Engineering Analysis with Boundary Elements, 50, 47-58 (2015).
H. J. Wu, Y. J. Liu, W. K. Jiang, and W. B. Lu, “A fast multipole boundary element method for three-dimensional half-space acoustic wave problems over an impedance plane,” International Journal of Computational Methods, 12, No. 1, 1350090 (2015).
F. Wu, G. R. Liu, G. Y. Li, Y. J. Liu, and Z. C. He, “A coupled ES-FEM and FM-BEM for structural acoustic problems,” Noise Control Engineering Journal, 62, No. 4, 196-209 (2014).
Y. J. Liu, Y. X. Li, and S. Huang, “A fast multipole boundary element method for solving two-dimensional thermoelasticity problems,” Computational Mechanics, 54, No. 3, 821-831 (2014).
Y. J. Liu and Y. X. Li, “Revisit of the equivalence of the displacement discontinuity method and boundary element method for solving crack problems,” Engineering Analysis with Boundary Elements, 47, 64–67 (2014).
Z. Guo, Y. J. Liu, H. Ma, and S. Huang, “A fast multipole boundary element method for modeling 2-D multiple crack problems with constant elements,” Engineering Analysis with Boundary Elements, 47, 1-9 (2014).
Y. J. Liu and Y. X. Li, “Slow convergence of the BEM with constant elements in solving beam bending problems,” Engineering Analysis with Boundary Elements, 39, 1-4 (2014).
Y. J. Liu, W. Ye and Y. Deng, “On the identities for elastostatic fundamental solution and nonuniqueness of the traction BIE solution for multi-connected domains,” ASME Journal of Applied Mechanics, 80, No. 5, 051012 (2013).
S. Huang and Y. J. Liu, “A fast multipole boundary element method for solving the thin plate bending problem,” Engineering Analysis with Boundary Elements, 37, No. 6, 967–976 (2013).
Y. J. Liu and S. Huang, “Identities for the fundamental solution of thin plate bending problems and the nonuniqueness of the hypersingular BIE solution for multi-connected domains,” Engineering Analysis with Boundary Elements, 37, No. 3, 579-584 (2013).
H. Ma, Z. Guo, M. Dhanasekar, C. Yan and Y. J. Liu, “Efficient solution of multiple cracks in great number using eigen COD boundary integral equations with iteration procedure,” Engineering Analysis with Boundary Elements, 37, No. 3, 487-500 (2013).
H. J. Wu, W. K. Jiang and Y. J. Liu, “Analyzing acoustic radiation modes of baffled plates with a fast multipole boundary element method,” ASME Journal of Vibration and Acoustics, 135, No. 1, 011007: 1-7 (2013).
H. J. Wu, Y. J. Liu and W. K. Jiang, “A low frequency fast multipole boundary element method based on analytical integration of the hypersingular integral for 3D acoustic problems,” Engineering Analysis with Boundary Elements, 37, No. 2, 309-318 (2013).
B. Kumar, D. I. Bylski-Austrow, and Y. J. Liu, “Finite element model of spinal hemiepiphysiodesis: effect of contact conditions, initial conditions, and growth,” Studies in Health Technology and Informatics, 176, 99-103 (2012).
H. J. Wu, Y. J. Liu and W. K. Jiang, “A fast multipole boundary element method for 3D multi-domain acoustic scattering problems based on the Burton-Miller formulation,” Engineering Analysis with Boundary Elements, 36, No. 5, 779–788 (2012).
H. J. Wu, Y. J. Liu and W. K. Jiang, “Analytical integration of the moments in the diagonal form fast multipole boundary element method for 3-D acoustic wave problems,” Engineering Analysis with Boundary Elements, 36, No. 2, 248–254 (2012).
H. J. Wu, W. K. Jiang and Y. J. Liu, “Diagonal form fast multipole boundary element method for 2D acoustic problems based on Burton-Miller boundary integral equation formulation and its applications,” Applied Mathematics and Mechanics, 32, No. 8, 981-996 (2011).
X. Y. Zhu, W. Q. Chen, Z. Y. Huang and Y. J. Liu, “A fast multipole boundary element method for 2D viscoelastic problems,” Engineering Analysis with Boundary Elements, 35, No. 2, 170-178 (2011).
X. Y. Zhu, W. Q. Chen, Z. Y. Huang and Y. J. Liu, “Fast multipole boundary element analysis of 2D viscoelastic composites with imperfect interfaces,” Science China: Technological Sciences, 53, No. 8, 2160–2171 (2010).
H. J. Wu, W. K. Jiang and Y. J. Liu, “Analysis of numerical integration error for Bessel integral identity in fast multipole method for 2D Helmholtz equation,” Journal of Shanghai Jiaotong University (Science), 15, No. 6, 690-693 (2010).
Y. J. Liu, “A new boundary meshfree method with distributed sources,” Engineering Analysis with Boundary Elements, 34, No. 11, 914-919 (2010).
M. S. Bapat and Y. J. Liu, “A new adaptive algorithm for the fast multipole boundary element method,” CMES: Computer Modeling in Engineering & Sciences, 58, No. 2, 161-184 (2010).
M. S. Bapat, L. Shen and Y. J. Liu, “Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems,” Engineering Analysis with Boundary Elements, 33, Nos. 8-9, 1113-1123 (2009).
Y. Yun, Z. Dong, N. Lee, Y. J. Liu, D. Xue, and et al., “Revolutionizing biodegradable metals“, Materials Today, 12, No. 10, 22-32 (2009).
Y. J. Liu, “A fast multipole boundary element method for 2-D multi-domain elastostatic problems based on a dual BIE formulation,” Computational Mechanics, 42, No. 5, 761-773 (2008).
Y. J. Liu, N. Nishimura, D. Qian, N. Adachi, Y. Otani and V. Mokashi, “A boundary element method for the analysis of CNT/polymer composites with a cohesive interface model based on molecular dynamics,” Engineering Analysis with Boundary Elements, 32, No. 4, 299–308 (2008).
Y. J. Liu, “A new fast multipole boundary element method for solving 2-D Stokes flow problems based on a dual BIE formulation,” Engineering Analysis with Boundary Elements, 32, No. 2, 139-151 (2008).
Y. J. Liu and L. Shen, “A dual BIE approach for large-scale modeling of 3-D electrostatic problems with the fast multipole boundary element method,” International Journal for Numerical Methods in Engineering, 71, No. 7, 837–855 (2007).
V. Mokashi, D. Qian and Y. J. Liu, “A study on the tensile response and fracture in carbon nanotube-based composites using molecular mechanics,” Composites Science and Technology, 67, Nos. 3-4, 530-540 (2007).
L. Shen and Y. J. Liu, “An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton-Miller formulation,” Computational Mechanics, 40, No. 3, 461-472 (2007).
L. Shen and Y. J. Liu, “An adaptive fast multipole boundary element method for three-dimensional potential problems,” Computational Mechanics, 39, No. 6, 681-691 (2007).
Y. J. Liu, “Dual BIE approaches for modeling electrostatic MEMS problems with thin beams and accelerated by the fast multipole method,” Engineering Analysis with Boundary Elements, 30, No. 11, 940-948 (2006).
Y. J. Liu and N. Nishimura, “The fast multipole boundary element method for potential problems: a tutorial,” Engineering Analysis with Boundary Elements, 30, No. 5, 371-381 (2006). (Corrected Figures 4 and 5)
Y. J. Liu, “A new fast multipole boundary element method for solving large-scale two-dimensional elastostatic problems,” International Journal for Numerical Methods in Engineering, 65, No. 6, 863-881 (2006).
Y. J. Liu, N. Nishimura and Z. H. Yao, “A fast multipole accelerated method of fundamental solutions for potential problems,” Engineering Analysis with Boundary Elements, 29, No. 11, 1016-1024 (2005).
Y. J. Liu, N. Nishimura and Y. Otani, “Large-scale modeling of carbon-nanotube composites by a fast multipole boundary element method,” Computational Materials Science, 34, No. 2, 173-187 (2005).
X. L. Chen and Y. J. Liu, “An advanced 3-D boundary element method for characterizations of composite materials,” Engineering Analysis with Boundary Elements, 29, No. 6, 513-523 (2005).
B. Keum and Y. J. Liu, “Analysis of 3-D frictional contact mechanics problems by a boundary element method,” Tsinghua Science and Technology, 10, No. 1, 16-29 (2005).
Y. J. Liu, N. Nishimura, Y. Otani, T. Takahashi, X. L. Chen and H. Munakata, “A fast boundary element method for the analysis of fiber-reinforced composites based on a rigid-inclusion model,” ASME Journal of Applied Mechanics, 72, No. 1, 115-128 (2005).
N. Nishimura and Y. J. Liu, “Thermal analysis of carbon-nanotube composites using a rigid-line inclusion model by the boundary integral equation method,” Computational Mechanics, 35, No. 1, 1-10 (2004).
Y. J. Liu and X. L. Chen, “Modeling of carbon nanotube-based composites by the boundary element method,” JASCOME Reviews, No. 2004-1, 45-57 (2004).
X. L. Chen and Y. J. Liu, “Square representative volume elements for evaluating the effective material properties of carbon nanotube-based composites,” Computational Materials Science, 29, No. 1, 1-11 (2004).
Y. J. Liu and X. L. Chen, “Evaluations of the effective material properties of carbon nanotube-based composites using a nanoscale representative volume element,” Mechanics of Materials, 35, No. 1-2, 69-81 (2003).
Y. J. Liu and X. L. Chen, “Continuum models of carbon nanotube-based composites using the boundary element method,” Electronic Journal of Boundary Elements, 1, No. 2, 316-335 (2003).
M. Nuggehally, Y. J. Liu, S. B. Chaudhari and P. Thampi, “An Internet-based computing platform for the boundary element method,” Advances in Engineering Software, 34, No. 5, 261-269 (2003).
D. Shi, J. Lian, P. He, L. M. Wang, W. J. van Oojj, M. Schulz, Y. J. Liu and D. B. Mast, “Plasma deposition of ultrathin polymer films on carbon nanotubes,” Applied Physics Letters, 81, No. 27, 5216-5218 (2002).
J. S. Yang and Y. J. Liu, “Boundary formulation and numerical analysis of elastic bodies with surface bonded piezoelectric films,” Smart Materials and Structures, 11, No. 2, 308-311 (2002).
Y. J. Liu and H. Fan, “Analysis of thin piezoelectric solids by the boundary element method,” Computer Methods in Applied Mechanics and Engineering, 191, No. 21-22, 2297-2315 (2002).
X. L. Chen and Y. J. Liu, “Thermal stress analysis of multi-layer thin films and coatings by an advanced boundary element method,” CMES: Computer Modeling in Engineering & Sciences, 2, No. 3, 337-349 (2001).
X. L. Chen and Y. J. Liu, “Multiple-cell modeling of fiber-reinforced composites with the presence of interphases using the boundary element method,” Computational Materials Science, 21, No. 1, 86-94 (2001).
Y. J. Liu and H. Fan, “On the conventional boundary integral equation formulation for piezoelectric solids with defects or of thin shapes,” Engineering Analysis with Boundary Elements, 25, No. 2, 77-91 (2001).
S. H. Chen, Y. J. Liu and X. Y. Dou, “A unified boundary element method for the analysis of sound and shell-like structure interactions. II. Efficient solution techniques,” The Journal of the Acoustical Society of America, 108, No. 6, 2738-2745, December (2000).
Y. J. Liu, “On the simple-solution method and non-singular nature of the BIE/BEM – A review and some new results,” Engineering Analysis with Boundary Elements (Special issue), 24, No. 10, 789-795 (2000).
Y. J. Liu and N. Xu, “Modeling of interface cracks in fiber-reinforced composites with the presence of interphases using the boundary element method,” Mechanics of Materials (Special issue: Mechanics & Mechanisms of Failure of Interfaces in Engineering Materials), 32, No. 12, 769-783 (2000).
Y. J. Liu, H. Fan and J. S. Yang, “Analysis of the shear stress transferred from a partially electroded piezoelectric actuator to an elastic substrate,” Smart Materials and Structures, 9, No. 4, 248-254 (2000).
J. F. Luo, Y. J. Liu and E. J. Berger, “Interfacial stress analysis for multi-coating systems using an advanced boundary element method,” Computational Mechanics, 24, No. 6, 448-455 (2000).
Y. J. Liu, N. Xu and J. F. Luo, “Modeling of interphases in fiber-reinforced composites under transverse loading using the boundary element method,” ASME Journal of Applied Mechanics, 67, No. 1, 41-49, March (2000).
Y. J. Liu and T. J. Rudolphi, “New identities for fundamental solutions and their applications to non-singular boundary element formulations,” Computational Mechanics, 24, No. 4, 286-292 (1999).
S. H. Chen and Y. J. Liu, “A unified boundary element method for the analysis of sound and shell-like structure interactions. I. Formulation and verification,” The Journal of the Acoustical Society of America, 106, No. 3, 1247-1254, September (1999).
Y. J. Liu and S. H. Chen, “A new form of the hypersingular boundary integral equation for 3-D acoustics and its implementation with C0 boundary elements,” Computer Methods in Applied Mechanics and Engineering, 173, No. 3-4, 375-386 (1999) (Special issue: Advances in Boundary Element Methods).
J. F. Luo, Y. J. Liu and E. J. Berger, “Analysis of two-dimensional thin structures (from micro- to nano-scales) using the boundary element method,” Computational Mechanics, 22, No. 5, 404-412 (1998).
Y. J. Liu, “Analysis of shell-like structures by the boundary element method based on 3-D elasticity: formulation and verification,” International Journal for Numerical Methods in Engineering, 41, No. 3, 541-558, February (1998).
Y. J. Liu and F. J. Rizzo, “Scattering of elastic waves from thin shapes in three dimensions using the composite boundary integral equation formulation,” The Journal of the Acoustical Society of America, 102, No. 2, 926-932, August (1997).
J. C. Chao, F. J. Rizzo, I. Elshafiey, Y. J. Liu, and et al., “General formulation for light scattering by a dielectric body near a perfectly conducting surface,” Journal of Optical Society of America A, 13, No. 2, 338-344 (1996).
J. C. Chao, Y. J. Liu, F. J. Rizzo, and et al., “Regularized integral equations and curvilinear boundary elements for electromagnetic wave scattering in three dimensions,” IEEE Trans. on Antennas and Propagat, 43, 1416-1422 (1995).
G. Krishnasamy, F. J. Rizzo, and Y. J. Liu, “Boundary integral equations for thin bodies,” International Journal for Numerical Methods in Engineering, 37, 107-121 (1994).
Y. J. Liu and F. J. Rizzo, “Hypersingular boundary integral equations for radiation and scattering of elastic waves in three dimensions,” Computer Methods in Applied Mechanics and Engineering, 107, 131-144 (1993).
Y. J. Liu and F. J. Rizzo, “A weakly-singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems,” Computer Methods in Applied Mechanics and Engineering, 96, 271-287 (1992).
G. Krishnasamy, F. J. Rizzo, and Y. J. Liu, “Some advances in boundary integral methods for wave-scattering from cracks,” Acta Mechanica, [Suppl], 3, 55-65 (1992).
Y. J. Liu and T. J. Rudolphi, “Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations,” Engineering Analysis with Boundary Elements, 8, 301-311 (1991).
Y. J. Liu, “Elastic stability analysis of thin plate by the boundary element method — a new formulation,” Engineering Analysis with Boundary Elements, 4, 160-164 (1987).
T. Q. Ye and Y. J. Liu, “Finite deflection analysis of elastic plate by the boundary element method,” Applied Mathematical Modelling, 9, 183-188 (1985).